M. Aprodu and V. Brˆınzˇ anescu Nagoya Math. J. Vol. 154 (1999), 111–122
MODULI SPACES OF VECTOR BUNDLES OVER RULED SURFACES ˇ MARIAN APRODU and VASILE BRˆINZANESCU Abstract. We study moduli spaces M (c1 , c2 , d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants c1 , c2 , d, r over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the nonemptiness of the space M (c1 , c2 , d, r) and we apply this result to the moduli spaces ML (c1 , c2 ) of stable bundles, where L is an ample line bundle on the ruled surface.
Introduction Let π : X → C be a ruled surface over a smooth algebraic curve C, defined over the complex number field C. Let f be a fibre of π. Let c1 ∈ Num(X) and c2 ∈ H 4 (X, Z) ∼ = Z be fixed. For any polarization L, denote the moduli space of rank-2 vector bundles stable with respect to L in the sense of Mumford-Takemoto by ML (c1 , c2 ). Stable 2-vector bundles over a ruled surface have been investigated by many authors; see, for example [T1], [T2], [H-S], [Q1]. Let us mention that Takemoto [T1] showed that there is no rank-2 vector bundle (having c1 .f even) stable with respect to every polarization L. In this paper we shall study algebraic 2-vector bundles over ruled surfaces, but we adopt another point of view: we shall study moduli spaces of (algebraic) 2-vector bundles over a ruled surface X, which are defined independent of any ample divisor (line bundle) on X, by taking into account the special geometry of a ruled surface (see [B], [B-St1], [B-St2] and also [Br1], [Br2], [W]). In Section 1 (put for the convenience of the reader) we present (see [B]) two numerical invariants d and r for a 2-vector bundle with fixed Chern classes c1 and c2 and we define the set M (c1 , c2 , d, r) of isomorphism classes of bundles with fixed invariants c1 , c2 , d, r. The integer d is given by the splitting of the bundle on the general fibre and the integer r is given by some normalization of the bundle. Recall that the set M (c1 , c2 , d, r) carries Received February 8, 1996.
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a natural structure of an algebraic variety (see [B], [B-St1], [B-St2]). In Section 2 we study uniform vector bundles and we prove the existence of algebraic vector bundles given by extensions of line bundles and which are not uniform. In Section 3 the main result gives necessary and sufficient conditions for the non-emptiness of the space M (c1 , c2 , d, r) and we apply this result to the moduli space of stable bundles ML (c1 , c2 ). §1. Moduli spaces of rank-2 vector bundles In this section we shall recall from ([B], [B-St1], [B-St2]) some basic notions and facts. The notations and the terminology are those of Hartshorne’s book [Ha]. Let C be a nonsingular curve of genus g over the complex number field and let π : X→C be a ruled surface over C. We shall write X ∼ = P(E) where E V is normalized. Let us denote by e the divisor on C corresponding to 2 E and by e = − deg(e). We fix a point p0 ∈ C and a fibre f0 = π −1 (p0 ) of X. Let C0 be a section of π such that OX (C0 ) ∼ = OP(E) (1). 2 ∼ Any element of Num(X) = H (X, Z) can be written aC0 + bf0 with a, b ∈ Z. We shall denote by OC (1) the invertible sheaf associated to the divisor p0 on C. If L is an element of Pic(C) we shall write L = OC (k) ⊗ L0 , where L0 ∈ Pic0 (C) and k = deg(L). We also denote by F (aC0 + bf0 ) = F ⊗ OX (a) ⊗ π ∗ OC (b) for any sheaf F on X and any a, b ∈ Z. Let E be an algebraic rank-2 vector bundle on X with fixed numerical Chern classes c1 = (α, β) ∈ H 2 (X, Z) ∼ = Z × Z, c2 = γ ∈ H 4 (X, Z) ∼ = Z, where α, β, γ ∈ Z. Since the fibres of π are isomorphic to P1 we can speak about the generic 0 splitting type of E and we have E|f ∼ = Of (d) ⊕ Of (d ) for a general fibre 0 0 f , where d ≤ d, d + d = α. The integer d is the first numerical invariant of E. The second numerical invariant is obtained by the following normalization: −r = inf{l| ∃L ∈ Pic(C), deg(L) = l, s.t. H 0 (X, E(−dC0 ) ⊗ π ∗ L) 6= 0}. We shall denote by M (α, β, γ, d, r) or M (c1 , c2 , d, r) or M the set of isomorphism classes of algebraic rank-2 vector bundles on X with fixed Chern classes c1 , c2 and invariants d and r. With these notations we have the following result (see [B]):
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Theorem 1. For every vector bundle E ∈ M (c1 , c2 , d, r) there exist L1 , L2 ∈ Pic0 (C) and Y ⊂ X a locally complete intersection of codimension 2 in X, or the empty set, such that E is given by an extension
(1)
0
0→OX (dC0 + rf0 )⊗π ∗ L2 →E→OX (d C0 + sf0 )⊗π ∗ L1 ⊗IY →0, 0
0
where c1 = (α, β) ∈ Z × Z, c2 = γ ∈ Z, d + d = α, d ≥ d , r + s = β, l(c1 , c2 , d, r) := γ + α(de − r) − βd + 2dr − d2 e = deg(Y ) ≥ 0. Remark. By applying Theorem 1 we can obtain the canonical extensions used in [Br1], [Br2]. 0 Indeed, let us suppose first that d > d . From the exact sequence (1) it follows that OC (r) ⊗ L2 ∼ = π∗ E(−dC0 ) so OX (rf0 ) ⊗ π ∗ L2 ∼ = π ∗ π∗ E(−dC0 ) and OX (dC0 + rf0 ) ⊗ π ∗ L2 ∼ = (π ∗ π∗ E(−dC0 ))(dC0 ). 0
If d = d then, by applying π∗ to the short exact sequence 0→OX (rf0 ) ⊗ π ∗ L2 →E(−dC0 )→OX (sf0 ) ⊗ π ∗ L1 ⊗ IY →0 it follows the exact sequence 0→OC (r) ⊗ L2 →π∗ E(−dC0 )→OC (s) ⊗ L1 ⊗ OC (−Z1 )→0, where Z1 is an effective divisor on C with the support π(Y ). With the notation Z = π −1 (Z1 ), by applying π ∗ (π is a flat morphism) we obtain the following commutative diagram with exact rows 0 - OX (rf0 ) ⊗ π ∗ L2 o 6id
- E(−dC0 ) 6ϕ
- OX (sf0 ) ⊗ π ∗ L1 ⊗ IY - 0 6ψ
0 - OX (rf0 ) ⊗ π ∗ L2 - π ∗ π∗ E(−dC0 ) - OX (sf0 ) ⊗ π ∗ L1 ⊗ IZ - 0
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From the injectivity of ψ we obtain the injectivity of ϕ. Because of OX (sf0 ) ⊗ π ∗ L1 ⊗ IY ⊂Z ∼ = Coker ψ ∼ = Coker ϕ we conclude. Recall that a set M of vector bundles on a C−scheme X is called bounded if there exists an algebraic C-scheme T and a vector bundle V on T × X such that every E ∈ M is isomorphic with Vt = V |t×X for some closed point t ∈ T (see [K]). For the next result see [B]: Theorem 2. The set M (c1 , c2 , d, r) is bounded. §2. Uniform bundles In what follows, we keep the notations from Section 1. Definition 3. A 2-vector bundle E is called an uniform bundle if the splitting type is preserved on all fibres of X. Theorem 1 allows us to give a criterion for uniformness. Lemma 4. Let f be a fibre of X and let us suppose that IY ∩f ⊂f ∼ = 0 Of (−n). Then E|f ∼ = Of (d + n) ⊕ Of (d − n). 0 0 Proof. We suppose that E|f ∼ = Of (a) ⊕ Of (a ), where a ≥ a . Then we have a surjective morphism 0
E|f →Of (d ) ⊗ IY ⊗ Of in virtue of Theorem 1. On the other hand, the restriction of the sequence 0→IY →OX →OY →0 to f gives a surjective morphism IY ⊗ Of →IY ∩f ⊂f ∼ = Of (−n). So, we obtain another surjective morphism 0
0
Of (a) ⊕ Of (a )→Of (d − n). 0
0
0
0
By using the inequalities a ≥ a , d ≥ d ≥ d − n and the equality a + a = 0 0 0 d + d = α it follows that a = d − n and a = d + n.
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Corollary 5. E is an uniform bundle if and only if l(c1 , c2 , d, r) = 0. By means of Corollary 5 the uniform bundles are given by extensions of line bundles. It is naturally to ask if the converse is true. Unfortunately, this question has a negative answer, as proved by the following Proposition 6. On the rational ruled surface Fe with e ≥ 1 there exist non-uniform bundles given by extensions of line bundles. For the proof we need some preparations. Let E be a 2-vector bundle given by an extension 0→F →E→G→0,
(2) 0
0
0
0
0
0
0
where F = OX (aC0 + r f0 ) ⊗ π ∗ L2 , G = OX (a C0 + s f0 ) ⊗ π ∗ L1 (L1 , L2 ∈ Pic0 (C)) are line bundles on X. By means of Theorem 1, E sits also in a 0 canonical extension (1). If a ≥ a then E is obviously uniform. Then, we 0 shall suppose that a < a . 0
Lemma 7. With the above notations we have d ≤ a . Proof. Indeed, by the restriction of the sequence (2) to a general fibre f we obtain a surjective morphism 0
0
Of (d) ⊕ Of (d )→Of (a ). 0
0
0
If d > a , then it follows that d = a which contradicts the inequalities 0 0 a < a , d ≥ d (a + a0 = d + d0 ). 0
Lemma 8. If d = a then E is uniform. Proof. Let f be a fibre of X such that the splitting type of E|f is different from the generic splitting type of E. According to Lemma 4 0 E|f ∼ = Of (d + n) ⊕ Of (d − n),
where n > 0. By the restriction of (2) to f we obtain a surjective morphism 0
Of (d + n) ⊕ Of (d − n)→Of (d). 0
Because of d + n > d it follows d − n = d, contradiction.
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ˇ M. APRODU AND V. BRˆıNZANESCU 0 Lemma 9. In the above hypotheses, if d = a , then E ∼ = F ⊕ G.
Proof. Let us observe that we can suppose, without loss of general0 ity, that a = 0 and r = 0 (by twisting the sequences (1) and (2) with 0 0 0 0 OX (−aC0 − r f0 )). Then, it follows that d = a = α > 0, s = β and d = 0. Therefore, the sequences (1) and (2) become: 0 6 0
OX (αC0 + βf0 ) ⊗ π ∗ L1 χ
6ϕ
ψ (10 ) 0 - OX (αC0 + rf0 ) ⊗ π ∗ L2 - E - OX (sf0 ) ⊗ π ∗ L1 ⊗ IY - 0 6 0
π ∗ L2 6
0 The computation of c2 (E) in (10 ) gives deg(Y ) = −αs. Moreover, by means of Lemma 8, deg(Y ) = 0, so s = 0 (we supposed α > 0). The homomorphism χ = ϕψ is non-zero, otherwise OX (αC0 + βf0 ) ⊂ 0 0 ∗ π (L2 ) (which would contradict the condition α > 0), so L2 = L1 and χ is the multiplication by a λ ∈ C∗ , and the assertion follows. In this moment, we are able to give the counter-example announced in Proposition 6. Proof of Proposition 6. Let G be OX (2C0 ) and let F be OX . Then: dim H 1 (G−1 ) = e + 1 6= 0. For E given by an extension ξ ∈ Ext1 (G, OX ), keeping the notations 0 0 from Section 1, we have d ≤ 2 (Lemma 7) , d ≥ d , d + d = 2 and r + s = 0. There are only two possibilities: 0 (a) d = 2 , d = 0, which implies E ∼ = OX ⊕ OX (2C0 ) (Lemma 9). 0 (b) d = d = 1 and, in this case, in the canonical extension (1) of E, we have
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0
deg(Y ) = dd e − ds − d r = e ≥ 1. By applying Corollary 5, all vector bundles given by non-zero extensions from Ext1 (G, OX ) are non-uniform. §3. Non-emptiness of moduli spaces For a rank-2 vector bundle E, we shall denote by dE and rE the invariants of E, when confusions may appear. Theorem 10. M (c1 , c2 , d, r) is non-empty if and only if l := l(c1 , c2 , d, r) ≥ 0 and one of the following conditions holds: (I) 2d > α or, (II) 2d = α, β − 2r ≤ g + l. Proof. We observe that if M 6= ∅ then, by means of Theorem 1, the elements of M lie among 2-vector bundles given by extensions of type (1). Therefore, we conclude that M 6= ∅ if and only if in the extensions of type (1) there are 2-vector bundles with dE = d and rE = r. It is clear that all the vector bundles given by an extension of type (1) have dE = d so we shall look for bundles with rE = r. We fix L1 , L2 ∈ Pic0 (C) and Y ⊂ X a locally complete intersection (or the empty set) and we denote 0
N1 = OX (d C0 + sf0 ) ⊗ π ∗ L1 N2 = OX (dC0 + rf0 ) ⊗ π ∗ L2 and l = deg(Y ). Consider the spectral sequence of terms E2p,q = H p (X, Extq (IY ⊗ N1 , N2 )) which converges to Extp+q (IY ⊗ N1 , N2 ). We have Ext0 (IY ⊗ N1 , N2 ) ∼ = OY . = N2 ⊗ N1−1 and Ext1 (IY ⊗ N1 , N2 ) ∼ But H 2 (X, N2 ⊗ N1−1 ) = 0 so the exact sequence of lower terms becomes 0→H 1 (X, N2 ⊗ N1−1 )→Ext1 (IY ⊗ N1 , N2 )→H 0 (Y, OY )→0.
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Now, by a result due to Serre (see [O-S-S], Chap.I, 5, [Se]), any element in the group Ext1 (IY ⊗ N1 , N2 ) which has an invertible image in H 0 (Y, OY ) defines an extension of the desired form with E a 2-vector bundle. We write the sequence (1) under the equivalent form (3)
00 0 ˜ ⊗ IY →0 0→OX →E(−dC0 ) ⊗ π ∗ L →OX ((d − d)C0 + (s − r)f0 ) ⊗ π ∗ (L)
˜ = L1 ⊗ L−1 , L = OC (−r) ⊗ L−1 and deg(L ) = −r. where L 2 2 From the definition, it follows r ≤ rE for every bundle E given by an extension (1). We distinguish three cases: 0 (I) d > d . In this case we shall prove that M is non-empty if and only if l ≥ 0. To do this we prove that all vector bundles from extension (1) have rE = r. 0 0 We verify that for all L ∈ Pic(C) with deg(L ) < 0 we have 00
00
00
0
H 0 (E(−dC0 ) ⊗ π ∗ (L ⊗ L )) = 0, 0
which is true because H 0 (L ) = 0 and H 0 (OX ((d − d)C0 + (s − r)f0 ) ⊗ π ∗ (L1 ⊗ L−1 2 ⊗ L ) ⊗ IY ) = 0. 0
0
(II) a◦ . d = d , r ≥ s. Then M is non-empty if and only if l ≥ 0. The proof 0 runs like in the first case with the remark deg(OC (s−r)⊗L1 ⊗L−1 2 ⊗L ) < 0. 0 (II) b◦ . d = d , r < s. Then M is non-empty if and only if l ≥ 0 and β − 2r ≤ g + l. Let us see first that the natural isomorphism 0
M (2d, β, γ, d, r)−→M (0, β, l, 0, r) E−→E(−dC0 ) 0
allows us to suppose d = d = 0. In this case, the sequence (3) becomes −1 ∗ 0→OX →E ⊗ OX (−rf0 ) ⊗ π ∗ L−1 2 →OX ((s − r)f0 ) ⊗ π (L1 ⊗ L2 ) ⊗ IY →0.
The definition of the second invariant implies that rE = r if and only 0 0 if E := π∗ E ⊗ OC (−rp0 ) ⊗ L−1 2 is normalised. E belong to an extension (4)
0
0→OC →E →L→0
where L = OC ((s − r)p0 ) ⊗ L1 ⊗ L−1 2 ⊗ OC (−Z1 ) with Z1 an effective divisor on C with support π(Y ) and card(Y ) ≤ deg(Z1 ) ≤ l = deg(Y ). 0 According to a result of Nagata ([N] or [Ha] Ex.V.2.5) , if E is normalised, then
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0
− deg(E ) = r − s + deg(Z1 ) ≥ −g which proves “only if” part of (II) b◦ . For “if” part we choose Y reduced, obtained by intersection between C0 and l distinct fibres of X. In this case, we have the following short exact sequence (5) 0→IZ →IY →IY ⊂Z →0 where Z1 = π(Y ) = p1 + · · · + pl , Y ⊂ Z = π −1 (Z1 ) = f1 + · · · + fl with fi distinct fibres, OZ = Of1 ⊕ · · · ⊕ Ofl , IY ⊂Z = Of1 (−1) ⊕ · · · ⊕ Ofl (−1) . So, the sequence (5) becomes 0→IZ →IY →Of1 (−1) ⊕ · · · ⊕ Ofl (−1)→0. Tensoring by KX ⊗ N2−1 ⊗ N1 and taking the long cohomology sequence we obtain an injective map: H 1 (KX ⊗ N2−1 ⊗ N1 ⊗ IZ )−→H 1 (KX ⊗ N2−1 ⊗ N1 ⊗ IY ). By dualizing, it follows that the natural map ϕ Ext1 (IY ⊗ N1 , N2 ) −→ Ext1 (IZ ⊗ N1 , N2 ) ∼ = Ext1 (L, OC )
is surjective, which shows that all bundles in (4) are coming from (1) by applying π∗ . According to [Ha] (Ex. V.2.5), there is a non-empty open set V ⊂ Ext1 (L, OC ) (don’t forget the condition s − r ≤ g + l !) such that all ξ ∈ V define normalised vector bundles on C. Now, in Ext1 (IY ⊗ N1 , N2 ) the set of vector bundles is a non-empty open set U . It is clear that ϕ−1 (V ) ∩ U 6= ∅ (being open sets in Zariski topology), so we conclude. §4. Moduli of stable bundles There is an interesting relation between the moduli spaces M (c1 , c2 , d, r) and the Qin’s sets Eζ (c1 , c2 ) (see [Q1], [Q2] for precised definitions). As in the proof of Theorem 10, case (I) we conclude that if ζ is a normalized class reprezenting a non-empty wall of type (c1 , c2 ) such that lζ (c1 , c2 ) > 0 then, for (2d−α, 2r −β) = ζ , Eζ (c1 , c2 ) and M (c1 , c2 , d, r) are coincident modulo a factor of Pic0 (C) (Qin workes with first Chern class c1 as an element in Pic(X)). This is a consequence of the following facts:
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(a) lζ (c1 , c2 ) = l(c1 , c2 , d, r) (b) condition ζ 2 < 0 implies 2d > α (c) in the case 2d > α the bundles L1 , L2 and the set Y from the sequence (1) are uniquely determined by E. (d) if l(c1 , c2 , d, r) > 0 then in the sequence (1) the bundles are given only by non-trivial extensions. In fact it is not hard to see that M (c1 , c2 , d, r) 6= ∅ iff Eζ (c1 , c2 ) 6= ∅ so, by means of Theorem 10, Eζ (c1 , c2 ) 6= ∅ if lζ (c1 , c2 ) > 0. But we have even more: Corollary 11. Let X be a ruled surface different from P1 × P1 and let C be a chamber of type (c1 , c2 ) different from Cf0 . Then the moduli space MC (c1 , c2 ) 6= ∅. Proof. From Theorem 1.3.3 in [Q2] it follows that F F MC (c1 , c2 ) = (MC1 (c1 , c2 ) − E(−ζ) (c1 , c2 )) Eζ (c1 , c2 ) , ζ
ζ
where C1 is the chamber lying above C and sharing with C a non-empty common wall W and ζ runs over all normalised classes representing W . By the above considerations, it follows that Eζ (c1 , c2 ) 6= ∅ if l(c1 , c2 , d, r) > 0. It remains the case l(c1 , c2 , d, r) = 0 and it will be sufficient to prove that h1 (X, N2 ⊗ N1−1 ) := dim H 1 (X, N2 ⊗ N1−1 ) > 0 (see the proof of Theorem 10). We have N2 ⊗ N1−1 = OX ((d − d0 )C0 + (r − s)f0 ) ⊗ π ∗ (L2 ⊗ L−1 1 ), where d − d0 = 2d − α = u and r − s = 2r − β = v. But ζ = uC0 + vf0 is a normalized class and this implies that u > 0 and v < 0 (see [Q1]). Because H 2 (X, N2 ⊗ N1−1 ) = 0, the Riemann-Roch Theorem gives the equality: χ = h0 (X, N2 ⊗N1−1 )−h1 (X, N2 ⊗N1−1 ) = 1−g+(1/2)((u+1)(2v−ue)+u(2−2g)).
But ζ 2 < 0 gives u(2v − ue) < 0; it follows 2v − ue < 0. If g ≥ 1, then obviously χ < 0. If g = 0, then e ≥ 0 and χ = 1 + v + (u/2)(2(v + 1) − e(u + 1)).
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If e ≥ 1, then χ < 0. For e = 0 we get X = P1 × P1 , which we excluded. Thus, in all cases χ < 0; it follows h1 (X, N2 ⊗ N1−1 ) > 0 and the proof is over. Remark. Let us suppose that X = P1 × P1 and that C is a chamber of type (c1 , c2 ) lying below a non-empty wall defined by a normalized class ζ = uC0 + vf0 with v ≤ −2. Then the same conclusion as in the above corollary holds. Indeed, in this case we have χ = (1 + v)(1 + u). Since v < −1, then again χ < 0. Acknowledgements. The second named author expresses his gratitude to the Max-Planck-Institut f¨ ur Mathematik Bonn for its hospitality during the final stage of this work.
References [B]
V. Brˆınzˇ anescu, Algebraic 2-vector bundles on ruled surfaces, Ann. Univ. Ferrara-Sez VII, Sc. Mat., XXXVII (1991), 55–64. [B-St1] V. Brˆınzˇ anescu and M. Stoia, Topologically trivial algebraic 2-vector bundles on ruled surfaces I, Rev. Roumaine Math. Pures Appl., 29 (1984), 661–673. , Topologically trivial algebraic 2-vector bundles on ruled surfaces II, In : [B-St2] Lect. Notes Math., 1056, Springer (1984). [Br1] J. E. Brossius, Rank-2 vector bundles on a ruled surface I, Math. Ann., 265 (1983), 155–168. [Br2] , Rank-2 vector bundles on a ruled surface II, Math. Ann., 266 (1984), 199–214. [Ha] R. Hartshorne, Algebraic Geometry, Graduate Texts in Math., 49, Springer, Berlin-Heidelberg, 1977. [H-S] H. J. Hoppe and H. Spindler, Modulr¨ aume stabiler 2-B¨ undel auf Regelfl¨ achen, Math. Ann., 249 (1980), 127–140. [K] S. Kleiman, Les th´ eor` emes de finitude pour les Founcteurs de Picard, In : Th´eories des intersections et th´ eor` eme de Riemann-Roch, SGA VI, Exp. XIII, Lect. Notes in Math., 225, Springer (1971), 616–666. [N] M. Nagata, On self-intersection Number of a section on ruled surface, Nagoya Math. J., 37 (1970), 191–196. [O-S-S] C. Okonek, M. Schneider and H. Spindler, Vector bundles on complex projective spaces, Birkh¨ auser, Basel Boston Stuttgart, 1980. [Q1] Z. Qin, Moduli spaces of stable rank-2 bundles on ruled surfaces, Invent. Math., 110 (1992), 615–626. [Q2] , Equivalence classes of polarizations and moduli spaces of sheaves, J. Diff. Geom., 37 (1993), 397–415.
122 [Se] [T1] [T2] [W]
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J. P. Serre, Sur les modules projectifs, S´em. Dubreil-Pisot 1960/1961 Exp. 2, Fac. Sci. Paris, 1963. F. Takemoto, Stable vector bundles on algebraic surfaces I, Nagoya Math. J., 47 (1972), 29–48. , Stable vector bundles on algebraic surfaces II, Nagoya Math. J., 52 (1973), 173–195. C. H. Walter, Components of the stack of torsion-free sheaves of rank-2 on ruled surfaces, Math. Ann., 301 (1995), 699–716.
Marian Aprodu Institute of Mathematics of the Romanian Academy P.O. BOX 1-764, RO-70700 Bucharest Romania
[email protected] Vasile Brˆınzˇ anescu Institute of Mathematics of the Romanian Academy P.O. BOX 1-764, RO-70700 Bucharest Romania
[email protected]
